arXiv:1411.2339v1 [hep-th] 10 Nov 2014
Spectral - integral representation of the photon polarization operator in a constant uniform magnetic field V.M. Katkov Budker Institute of Nuclear Physics, Novosibirsk, 630090, Russia e-mail:
[email protected]
Abstract The polarization operator in a constant and homogeneous magnetic field of arbitrary strength is investigated on mass shell. The calculations are carried out at all photon energies higher the pair creation threshold as well as lower this threshold. The general formula for the effective mass of the photon with given polarization has been obtained being useful for an analysis of the problem under consideration as well as at a numerical work. Approximate expressions for strong and weak fields H, compared with the critical field H0 = 4.41 · 1013 G, have been found. Depending on H/H0 we consider the pure quantum region of photon energy, where particles are created on lower Landau levels or created not at all. Also the energy region of large level numbers is considered where the quasiclassical approximation is valid.
1. The study of QED processes in a strong magnetic field close to and exceeding the critical field strength H0 = m2 /e = 4, 41 · 1013 G (the system of units ~ = c = 1 is used ) is stimulated essentially by the existence of very strong magnetic fields in nature. It is universally recognized the magnetic field of neutron stars (pulsars) run up H ∼ 1011 ÷ 1013 G [1]. These values of field strength gives the rotating magnetic dipole model, in which the pulsar loses rotational energy through the magnetic dipole radiation. The prediction of this model is in quite good agreement with the observed radiation from pulsars in the radio frequency region. There are around some thousand radio pulsars. Another class of neutron stars, now referred to as magnetars [2], was discovered on examination of the observed radiation at x-ray and γ-ray energies and may possess even stronger surface magnetic fields H ∼ 1014 ÷ 1015 G. The photon propagation in these fields and the dispersive properties of the space region with magnetic is of very much interest. This propagation accompanied by the photon conversion into a pair of charged particles when the transverse photon momentum is larger than the process threshold value k⊥ > 2m. When the field change is small on 1
the characteristic length of process formation (for example, when this length is smaller then the scale of heterogeneity of the neutron star magnetic field), the consideration can be realized in the constant field approximation. In 1971 Adler [3] had calculated the photon polarization operator in a magnetic field using the proper-time technique developed by Schwinger [4]. In the same year Batalin and Shabad [5] had calculated this operator in an electromagnetic field using the Green function found by Schwinger [4]. In 1975 the contribution of charged-particles loop in an electromagnetic field with n external photon lines had been calculated in [6]. For n = 2 the explicit expressions for the contribution of scalar and spinor particles to the polarization operator of photon were given in this work. For the contribution of spinor particles obtained expressions coincide with the result of [5], but another form is used. The polarization operator in a constant magnetic field has been investigated well enough in the energy region lower and near the pair creation threshold (see, for example, the papers [7, 8, 9] and the bibliography cited there. In the present paper we consider in detail the polarization operator on mass shell ( k 2 = 0, the metric ab = a0 b0 − ab is used ) at arbitrary value of the photon energy and magnetic field strength. The restriction of our consideration is only the applicability of the perturbation theory over the electromagnetic interaction constant α [10]. 2. Our analysis is based on the general expression for the contribution of spinor particles to the polarization operator obtained in a diagonal form in [6] (see Eqs. (3.19), (3.33)). For the case of pure magnetic field we have in a covariant form the following expression Πµν = −
X
i=2,3
κi βiµ βiν ,
βi βj = − δij ,
p β2µ = (F ∗ k)µ / −(F ∗ k)2 ,
F F ∗ = 0,
βi k = 0;
p β3µ = (F k)µ / −(F ∗ k)2 ,
F 2 = F µν Fµν = 2(H 2 − E 2 ) > 0,
(1)
(2)
where F µν − the electromagnetic field tensor , F ∗µν − dual tensor, k µ − the photon momentum, (F k)µ = F µν kν , κi = Here
R α 2 R1 ∞−i0 m r dv fi (v, x) exp[iψ(v, x)]dx. π −1 0
cos(vx) − cos x cos(vx) cos x sin(vx) − , +v sin x sin3 x sin2 x cos(vx) cos x sin(vx) − (1 − v 2 ) cot x, f3 (v, x) = −v 2 sin x sin x 1 cos x − cos(vx) ψ(v, x) = + [r(1 − v 2 ) − 1]x ; 2r µ sin x
(3)
f2 (v, x) = 2
r = −(F ∗ k)2 /2m2 F 2 ,
2
µ2 = F 2 /2H02 .
(4) (5)
The real part of κi determines the refractive index ni of the photon with polarization ei = βi : Reκi . (6) 2ω 2 At r > 1 the proper value of polarization operator κi includes the imaginary part which determines the probability per unit length of pair production by photon with the polarization βi : ni = 1 −
1 (7) Wi = − Imκi ω For r < 1 the integration counter over x in Eq. (3) may be turn to the lower semiaxis (x → −ix), then the value κi becomes real in the explicit form. 3. As well as in our work [11] (see Appendix A) we present the effective mass in the form of R1 ∞−i0 R r fi (v, x) exp[iψ(v, x)]dx, κi = αm2 Ti ; Ti = dv π −1 0 ∞ X R1 ∞−i0 R δn0 (n) (n) (n) 1− Ti = Ti ; Ti = dv Fi (v, x) exp[ian (v)x]dx, 2 −1 0 n=0
(8) (9)
where (n)
F1
(n)
F2
(n)
i 2vn (Jn+1 (t) − Jn−1 (t)) − cot xJn (t) , sin x z i 4 (n) b cot x − Jn (t) − F1 , = (−i)n exp(iz cot x) z sin2 x = (−i)n exp(iz cot x)
(n)
− 2(−i)n exp(iz cot x)(1 − v 2 ) cot xJn (t); 2r z 1 . an (v) = nv − b, b = (1 − r(1 − v 2 )), z = , t = µ µ sin x F3
= F1
(10) (11)
Let’s note that at x → −i∞ the asymptotic of the Bessel function Jn (t) is (iz)n −n|x| e , (12) n! and under the condition an (v) < n, the integration counter over x in Eq. (9) (n) can be unrolled to the lower semiaxis. Then Ti becomes real in the explicit form. (n) (n) The functions Fi (v, x) are periodical over x. So one can present Ti as Jn (t) ≃ Jn (2ize−|x|) ≃
3
(n)
Ti
=
R1
−1
∞ 2π X R (n) dv Fi (v, x) exp[ian (v)x]dx exp[2πikan (v)] 0
k=0
2π R (n) dv Fi (v, x) exp[ian (v)x]dx. = −1 1 − exp[2πian (v)] + i0 0
R1
(13)
Using the expression P 1 = − iπδ(1 − exp[2πian (v)]), 1 − exp[2πian (v)] + i0 1 − exp[2πian (v)]
(14)
taking into account the above notation − iπδ(1 − exp[2πian (v)]) X 1X = −iπ δ(1 − exp[2πi(an (v) − m)]) → δ(an (v) − m). 2 m
(15)
m≥n
(n)
and allowing for Fi
(n)
Ti
(n)
(v, x + π) = (−1)n Fi
i R1 dv = (−1)n P 2 −1 sin(πan (v))
Zπ
(n)
Fi
(v, x), we have
(v, x) exp[ian (v)x]dx
−π
+
m=n Xmax X m≥n v1,2
1 + (−1)m+n ϑ(g(n, m, r)) 2|a′n (v)|
Zπ
(n)
Fi
(v1,2 , x) exp[imx]dx, (16)
−π
where g(n, m, r) = r2 − (1 + mµ)r + n2 µ2 /4, nµ 1 √ 2√ v1,2 = ± g, a′n (v) = g; 2r r µ √ 2(r − r) nmax = [d(r)], d(r) = . µ Here [d] is the integer part of d. (n) Bringing out the distinction in the explicit form we present Ti as
4
(17) (18)
(n)
Ti
(nr)
Ti
(nr)
= Ti
(ns)
+ Ti
;
i R1 = (−1)n P dv 2 −1
(19) Zπ
(n)
[Fi
(v, x)
−π
(ns)
m=n Xmax X
exp[imx] (−1)m (n) Fi (v1,2 , x) ]dx, (20) π a n (v) − m m≥n v1,2 √ √ m=n Xmax X µπ 2 −g 1 2 −g = + arctan arctan √ 1− 2 g π 2r − µn 2r + µn v
− Ti
exp[ian (v)x] sin(πan (v))
m≥n
×
Zπ
(n)
Fi
1,2
(v1,2 , x) exp[imx]dx.
(21)
0
(nr)
is singularity- free, and for n > nmax the Here the regularized function Ti (n) integration counter in Ti can be unrolled to the lower semiaxis. . After that we present Ti in the form ∞ X
Ti =
(n) Ti
n>nmax
=
R1
+
nX max
(n) Ti
=
n=0
Ti −
R∞ dv {Fi (v, x) exp[−χ(v, x)]
0 −1 nX max
+i
(n)
Fi
n=0
nX max
(n) Ti
n=0
(v, −ix) exp[an (v)x]}dx +
!
+
nX max
(n)
Ti
(22)
n=0
(23) nX max
(n)
Ti
.
n=0
Here the functions Fi (v, x), χ(v, x), χ00 (v, x) have a form cosh x − cosh(vx) 1 2 F2 (v, x) = − cosh(vx) + v sinh(vx) coth x , (24) sinh x sinh2 x cosh x sinh(vx) cosh(vx) −v F3 (v, x) = − (1 − v 2 ) coth x; (25) sinh x sinh2 x cosh x − cosh(vx) 1 2r + (rv 2 − r + 1)x , µ sinh x 1 2r + (rv 2 − r + 1)x . χ00 (v, x) = µ χ(v, x) =
(26) (27)
and an (v) is given by (11) (ns) (only there the imaginary The integrals over x in the expression for Ti terms are contained) have been calculated in Appendix A [11]. Along with 5
integers m and n we use also l = (m + n)/2 and k = (m − n)/2 which are straight the level numbers p p rlk = (ε(l) + ε(k))2 /4m2 , ε(l) = m2 + 2eHl = m 1 + 2µl. (28)
We have
nmax X δn0 ζ n k! rX (ns) = −iαm2 µe−ζ 1− Ti (2 − δn0 ) √ = αm π n=0 2 gl! n,m √ √ 2 −g 2 −g 1 arctan Di ; (29) + arctan × 1− π 2r − µn 2r + µn mµ n2 µ2 n−1 D2 = F + 2µlϑ(k − 1) 2Ln+1 (ζ) − Lnk (ζ)Lnk−1 (ζ) , − k−1 (ζ)Lk 2 4r mµ n2 µ2 F + 2µlϑ(k − 1)Lnk (ζ)Lnk−1 (ζ), − D3 = 1 + 2 4r 2 l 2r 2 F = [Lnk (ζ)] + ϑ(k − 1) Lnk−1 (ζ) , ζ = , (30) k µ κsi
2
where Lnk (ζ) is the generalized Laguerre polynomial. At µ 1 Lnk (ζ) ≃ ζ k /k!,
D3 ≃ [Lnk (ζ)]2 ,
D2 ≃ mµD3 /2.
(32)
Eq. (31) coincides with Eq. (17) in [12]. Note that for g > 0 the imaginary part of Eq. (29) gives in addition the partial probability of level population by created particles (see [11]). At µ & 1, |r − 1|
